Delta-sigma modulation is widespread in both radio-frequency (RF) and analog/digital conversion fields. A delta-sigma modulator (DSM) uses noise-shaping and oversampling features to push noise and spurs due to device mismatch out of the frequency band of interest, where they can be easily filtered out. The higher the modulator frequency, the higher the amount of noise or spurs that may be filtered. For instance, a DSM may be applied to digital-to-analog converters (DACs) when a high number of bits Nb, e.g., greater than eight, is desired. In this case, the desired device matching (limited to a few parts percent) may not be enough to ensure desired linearity. A DSM may provide an output with a lower number of bits, e.g., two or three, at a higher sampling rate but having Nb-bit accuracy.
A common topology of a DSM is a multi-stage noise shaping (MASH) modulator. A second-order MASH DSM includes two cascaded accumulators with two output bits derived by combining a carry output of each accumulator to achieve the desired noise shaping function. As each accumulator is equivalent to a first order modulator, any order modulator with a MASH topology may be implemented by cascading accumulators and modifying a logical network combining all the carry outputs. Thus, a MASH modulator may be a cascade of first-order modulators, which may enable modular design without stability issues.